Question

Evaluate the expression without using a calculator.
log Subscript 2 Baseline 512
Question content area bottom
Part 1
log Subscript 2 Baseline 512equals
  
enter your response here

Answer 1

To evaluate \( \log_2 512 \), we need to express 512 as a power of 2.

First, we can start by breaking down 512:

\[
512 = 2^9
\]

This is because:

\[
2^1 = 2 \\
2^2 = 4 \\
2^3 = 8 \\
2^4 = 16 \\
2^5 = 32 \\
2^6 = 64 \\
2^7 = 128 \\
2^8 = 256 \\
2^9 = 512
\]

Now that we've established \( 512 = 2^9 \), we can use the properties of logarithms to simplify \( \log_2 512 \):

\[
\log_2 512 = \log_2 (2^9)
\]

Using the logarithmic identity \( \log_b (b^x) = x \), we have:

\[
\log_2 (2^9) = 9
\]

Thus, the value of \( \log_2 512 \) is:

\[
\log_2 512 = 9
\]

So the final answer is:

\[
\boxed{9}
\]